This 30 credit M-Level module brings together two areas in theoretical physics: one covering topics in mathematical physics such as vector analysis and complex variable theory; the other on nonlinear physics. Aspects of the material taught in these two lecture based courses is then investigated in depth via extensive coursework problems.

PH2M –Mathematical, Quantum and Computational Physics

To provide a sound grounding in the basic theorems, methods and applications of functions of a complex variable, a range of advanced integration techniques and theorems and the theory of nonlinear dynamical systems. To then encourage the in depth investigation of aspects of this material through extended coursework problems.

Students will have a solid understanding of derivatives and integrals of vector functions and appreciate their importance in almost every area of applied mathematics. They will appreciate the meaning of and be able to test a function for analyticity and also identify and classify poles and other singular points of functions. Students should be familiar with methods for performing real and complex variable integrals by complex contour integration.

Students will further appreciate the implications of nonlinearity in and learn to analyze and classify nonlinear dynamics by identifying periodic, quasi-periodic or chaotic behaviour.

On completion of the problem solving part of this multiple module the student should have an in-depth knowledge of aspects of theoretical physics covered in the lecture courses.

Mathematical Methods

Vector multiplication and differentiation, vector fields; directional derivative (grad); line and surface integrals; Green’s Theorem in the plane; the divergence and the Divergence (Gauss’s) Theorem; the curl and Stoke’s Theorem; examples. (8 lectures).

Functions of a complex variable, continuity, differentiability, the Cauchy-Reimann conditions, relationship to Laplace problems in 2dimensions, analyticity, singularities, poles.  Complex integration, Cauchy's theorem.  Residues and the residue theorem. Taylor and Laurent series. Application of residue theorem for evaluation of integrals. Examples. (12 lectures).

Nonlinear Physics:

Linear and nonlinear dynamics. Fixpoints, strange attractors and Chaos. Lyapunov-exponents and fractal dimensions. Local bifurcation theory. Principe of Chaos  control. Examples from physics, engineering and biology.

Problem Solving in Physics

This module should challenge MPhys students with more extensive problems than undergraduates encounter in formal written examinations.

There will be two problems set from each of the two lecture courses (Mathematical Methods and Nonlinear Physics).  Each student will tackle all the problems. Two will be near the beginning of semester and then two half way through.

This component of the multiple module is run on an individual tutorial basis, students meeting for sessions as necessary with the two course convenors.  Both initiative and originality will be expected from students in their approach to the problems.

40 hours of lectures (20 each for the two courses on Mathematical Methods and Nonlinear Physics) plus open ended open book study and tutorials for the coursework problems.

i.                 M L Boas, Mathematical Methods in the Physical Sciences, Third Edition,Wiley 2006

ii.               G Arfken and H B Weber, Mathematical Methods for Physicists, Academic Press, 2005

iii.       S H Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology,           Chemistry and Engineering, Perseus Books, 2000